Classic Croque Madame: The Ultimate Ham and Cheese with an Egg

Preheat oven to 375°F (190°C). Melt 1 tablespoon of butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly, to create a roux.

Gradually whisk in the warm milk until smooth. Bring to a simmer, then reduce heat to low and cook for 3-5 minutes, stirring occasionally, until the sauce thickens. Stir in the Dijon mustard, salt, pepper, and nutmeg. Remove from heat.

Lightly toast the bread slices. Spread a thin layer of the béchamel sauce on one side of each bread slice. On two of the slices, layer 2 slices of ham and half of the shredded Gruyère cheese.

Top with the remaining two bread slices, béchamel-side down, to form two sandwiches. Spread the remaining béchamel sauce generously over the top of each sandwich and sprinkle with the remaining Gruyère cheese.

Place the sandwiches on a baking sheet and bake for 10-12 minutes, or until the cheese is melted and bubbly and the edges are golden brown.

While the sandwiches are baking, melt the remaining 1 tablespoon of butter in a non-stick skillet over medium heat. Crack the two eggs into the skillet and fry them to your desired doneness, typically 2-3 minutes for a runny yolk.

Carefully transfer each baked sandwich to a plate and top with a fried egg. Serve immediately.